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Register a new userApollonian Packings and Kac-Moody Root Systems
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Authors
Ian Whitehead
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We study Apollonian circle packings in relation to a certain rank 4 indefinite Kac-Moody root system $Phi$. We introduce the generating function $Z(mathbf{s})$ of a packing, an exponential series in four variables with an Apollonian symmetry group, which relates to Weyl-Kac characters of $Phi$. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of $Phi$, with automorphic Weyl denominators, we express $Z(mathbf{s})$ in terms of Jacobi theta functions and the Siegel modular form $Delta_5$. We also show that the domain of convergence of $Z(mathbf{s})$ is the Tits cone of $Phi$, and discover that this domain inherits the intricate geometric structure of Apollonian packings.
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In this note, we provide a complete description of the closed sets of real roots in a Kac-Moody root system.
Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorphic continuation.
Let R be a finitely generated commutative ring with 1, let A be an indecomposable 2-spherical generalized Cartan matrix of size at least 2 and M=M(A) the largest absolute value of a non-diagonal entry of A. We prove that there exists an integer n=n(A) such that the Kac-Moody group G_A(R) has property (T) whenever R has no proper ideals of index less than n and all positive integers less than or equal to M are invertible in R.
We study a correction factor for Kac-Moody root systems which arises in the theory of $p$-adic Kac-Moody groups. In affine type, this factor is known, and its explicit computation is the content of the Macdonald constant term conjecture. The data of the correction factor can be encoded as a collection of polynomials $m_lambda in mathbb{Z}[t]$ indexed by positive imaginary roots $lambda$. At $t=0$ these polynomials evaluate to the root multiplicities, so we consider $m_lambda$ to be a $t$-deformation of $mathrm{mult} (lambda)$. We generalize the Peterson algorithm and the Berman-Moody formula for root multiplicities to compute $m_lambda$. As a consequence we deduce fundamental properties of $m_lambda$.
We initiate an investigation of lattices in a new class of locally compact groups, so called locally pro-$p$-complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well-behaved: under mild assumptions, a cocompact lattice in this completion contains no elements of order $p$. This statement is still an open question for the Caprace-Remy-Ronan completion. Using this, modulo results of Capdeboscq and Thomas, we classify edge-transitive cocompact lattices and describe a cocompact lattice of minimal covolume.
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